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Palindromic Sums of
Squares of Consecutive Integers
rood Sums of Cubes rood Sums of Primes rood Sums of Powers rood comments



Introduction

Palindromic numbers are numbers which read the same from

 p_right left to right (forwards) as from the right to left (backwards) p_left
Here are a few random examples : 535, 3773, 246191642



Palindromic Sums of Squares

A palindromic coincidence occurred with Five Consecutive Integers or is there a pattern ?

99 + 100 + 101 + 102 + 103 = 505
992 + 1002 + 1012 + 1022 + 1032 = 51015

10099 + 10100 + 10101 + 10102 + 10103 = 50505
100992 + 101002 + 101012 + 101022 + 101032 = 510151015

Two equations like 'number'-pregnancy. The children are the numbers 1981 and 699.
112 + 122 + 132 = 434
1198112 + 1198122 + 1198132 = 43064746034

372 + 382 + 392 = 4334
369972 + 369982 + 369992 = 4106556014


Messages

[ August 1, 2005 ]
Jean-Claude Rosa (email) ! [ goto entry ]

Cher Patrick,

Voici les numéros 38 et 39 qui hélas ne sont pas premiers
(le numéro 38 a 5 facteurs premiers mais le numéro 39 est
un semi-prime -un "vrai"- avec un facteur de 15 chiffres et
un autre de 17 chiffres ):

numéro 38 :
1682059335368470748635339502861 = 29 * 109 * 35935429 * 206697457 * 71640539617
917076696729469^2 + 917076696729470^2

numéro 39:
3167016920841776771480296107613 = 113665666219493 * 27862564186498841
1258375325735882^2 + 1258375325735883^2

Voilà c'est tout pour les palindromes de la forme 1... ...1 ou
3... ...3 de 31 chiffres. J'ai commencé à chercher ceux de la
forme 5... ...5 .
Pour espèrer trouver un palprime il faut donc s'attaquer aux
nombres de 33 chiffres ( vraiment dur, dur ! )

A bientôt.
JCR

[ September 6, 2005 ]
Jean-Claude Rosa (email) ! [ goto entry ]

Cher Patrick ,

Voici le numéro 40 :

5450871426137276727316241780545 =
1650889370330016^2 + 1650889370330017^2

C'est le dernier des palindromes de 31 chiffres !
J'ai donc commencé la recherche des palindromes de
33 chiffres mais cela risque de prendre pas mal de temps...

A bientôt.

Kind regards.
JCR

[ September 15, 2006 ]
Jean-Claude Rosa (email) ! [ goto entry ]

Cher Patrick,
Ca y est !! J'ai enfin trouvé le numéro 41 le voici :

316370934175751979157571439073613
=12577180410882082^2 + 12577180410882083^2

mais hélas il n'est pas premier (4 facteurs ) et comme j'ai fini l'étude
des palindromes de la forme 31...13 pour trouver la cinquième
solution du puzzle 14 il faudra étudier les nombres de 35 chiffres...
je ne sais pas si j'aurai le temps et le courage !
Je vais quand même essayer de finir l'étude des palindromes de 33 chiffres
de la forme 5.....5.

JCR

[ October 14, 2006 ]
Jean-Claude Rosa (email) ! [ goto entry ]

Cher Patrick,

... et je viens de trouver le numéro 42.
Le voici:
501263304966749757947669403362105
=15831350305118476^2 + 15831350305118477^2

Voilà, c'est tout pour aujourd'hui...
j'espère arriver un jour au numéro 50 ;-)

JCR

[ May 21, 2007 ]
Jean-Claude Rosa (email) ! [ goto entry ]

Cher Patrick,
Ca y est ! J'ai enfin trouvé le numéro 43.
Le voici :
562318868215014101410512868813265 =
16767809460615511^2 + 16767809460615512^2

Amitiés.
JCR

[ January 12, 2008 ]
Jean-Claude Rosa (email) ! [ goto entry ]

Cher Patrick,
... avec un peu de retard voici le numéro 44 :
588186547187469040964781745681885 =
17149147897016181^2 + 17149147897016182^2

Best regards.
JCR

[ February 17, 2009 ]
Jean-Claude Rosa (email) ! [ goto entry ]

Cher Patrick,
Voilà dejà un an que je n'ai pas eu de tes nouvelles
( le temps passe... ) mais j'attendais de trouver le numéro 45
pour t'écrire. Ca y est ! mais hélas il n'est pas premier
( petite consolation c'est un semi-prime !). Le voici :
10489844562990321812309926544898401 =
72421835667809200^2 + 72421835667809201^2

A bientôt.
JCR


Sources Revealed

Huen Y.K. from Singapore developed a general generating function for palindromic sums and products
of consecutive integers using concise programcode written for Macsyma 2.2.1.
Global Generating Function For Palindromic Sums and Products of Consecutive Integers.

See also Powers of Consecutives Summing to Palindromes and WONplate 145



Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online :
Neil Sloane's Integer Sequences
separator
The regular numbers of form n2 + (n+1)2 [sums of two squared consecutives] :
%N Centered square numbers: 2n(n-1)+1. under A001844.
The subsets in relation to the palindromic and prime numbers of above form :
%N n^2 + (n+1)^2 is palindromic. under A027571
%N Palindromes of form n^2 + (n+1)^2. under A027572.
%N n^2 + (n+1)^2 is prime. under A027861
%N Primes of form n^2 + (n+1)^2. under A027862.
separator
The regular numbers of form n2 + (n+1)2 + (n+2)2 [sums of three squared consecutives] :
%N Points on surface of square pyramid: 3*n^2 + 2. under A005918.
The subsets in relation to the palindromic and prime numbers of above form :
%N n^2 + (n+1)^2 + (n+2)^2 is palindromic. under A027573
%N Palindromes of form n^2 + (n+1)^2 + (n+2)^2. under A027574.
%N n^2 + (n+1)^2 + (n+2)^2 is prime. under A027863
%N Primes of form n^2 + (n+1)^2 + (n+2)^2. under A027864.
separator
The regular numbers of form n2 + (n+1)2 + (n+2)2 + (n+3)2 [sums of four squared consecutives] :
%N n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2. under A027575.
The subsets in relation to the palindromic numbers of above form :
%N n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 is palindromic. under A027576
%N Palindromes of form n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2. under A027577.
separator
The regular numbers of form n2 + (n+1)2 + (n+2)2 + (n+3)2 + (n+4)2 [sums of five squared consecutives] :
%N n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2. under A027578.
The subsets in relation to the palindromic numbers of above form :
%N n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2 is palindromic. under A027579
%N Palindromes of form n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2. under A027580.
separator
The regular numbers of form n2 + (n+1)2 + (n+2)2 + (n+3)2 + (n+4)2 + (n+5)2 :
%N n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2 + (n+5)^2. under A027865.
The subsets in relation to the prime numbers of above form :
%N n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2 + (n+5)^2 is prime. under A027866
%N Primes of form n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2 + (n+5)^2. under A027867.
separator
Click here to view some of the author's [P. De Geest] entries to the table.
Click here to view some entries to the table about palindromes.

A simple substitution of n with (m–1) will proof that n^2 + (n+1)^2 + (n+2)^2 equals 3*n^2 + 2 :

n^2 + (n+1)^2 + (n+2)^2 =
(m–1)^2 + ((m–1)+1)^2 + ((m–1)+2)^2 =
(m^2–2m+1) + (m^2) + (m^2+2m+1) =
m^2 + 1 + m^2 + m^2 + 1 =
3*m^2 + 2 QED
And also that n^2 + (n+1)^2 equals 2n(n–1)+1 :
n^2 + (n+1)^2 =
(m–1)^2 + ((m–1)+1)^2 =
m^2–2m+1 + m^2 =
2m^2 – 2m + 1 =
2m(m–1) + 1 QED


The Table



Index Nr Base Square Expression 
Palindromic Sums of Squares of Consecutive Integers Length 
   
Sums of Squares of FIVE Consecutive Integers
Searched for palindromes upto length 17.
5 10.0992 + 10.1002 + 10.1012 + 10.1022 + 10.10325
510.151.0159
4 3.2072 + 3.2082 + 3.2092 + 3.2102 + 3.21124
51.488.4158
3 3312 + 3322 + 3332 + 3342 + 33523
554.4556
2 992 + 1002 + 1012 + 1022 + 10322 - 3
51.0155
1 12 + 22 + 32 + 42 + 521
552
   
Sums of Squares of FOUR Consecutive Integers
(Palindromes are of odd-length only)
Searched upto length 17.
6 102.025.3002 + 102.025.3012 + 102.025.3022 + 102.025.30329
41.636.648.584.663.61417
5 101.740.1722 + 101.740.1732 + 101.740.1742 + 101.740.17529
41.404.251.615.240.41417
4 12.309.5982 + 12.309.5992 + 12.309.6002 + 12.309.60128
606.104.959.401.60615
3 10.460.1372 + 10.460.1382 + 10.460.1392 + 10.460.14028
437.657.989.756.73415
2 10.1722 + 10.1732 + 10.1742 + 10.17525
414.000.4149
1 1002 + 1012 + 1022 + 10323
41.2145
   
Sums of Squares of THREE Consecutive Integers
Submissions from Jean Claude Rosa (email)
Searched for palindromes upto length 21.
Index 22 and 23 [ August 28, 2002 ]
Index 24 up to 28 [ August 30, 2002 ]
28 17.552.348.1962 + 17.552.348.1972 + 17.552.348.198211
924.254.781.686.187.452.42921
27 16.028.474.3442 + 16.028.474.3452 + 16.028.474.346211
770.735.969.484.969.537.07721
26 12.900.321.3912 + 12.900.321.3922 + 12.900.321.393211
499.254.876.050.678.452.99421
25 8.227.749.4892 + 8.227.749.4902 + 8.227.749.491210
203.087.585.010.585.780.30221
24 5.602.945.2422 + 5.602.945.2432 + 5.602.945.244210
94.178.986.188.168.987.14920
23 828.223.2492 + 828.223.2502 + 828.223.25129
2.057.861.255.521.687.50219
22 441.564.3802 + 441.564.3812 + 441.564.38229
584.937.307.703.739.48518
21 40.439.5572 + 40.439.5582 + 40.439.55928
4.906.073.553.706.09416
20 17.909.2822 + 17.909.2832 + 17.909.28428
962.227.252.722.26915
19 8.211.8792 + 8.211.8802 + 8.211.88127
202.304.919.403.20215
18 4.427.7802 + 4.427.7812 + 4.427.78227
58.815.733.751.88514
17 1.775.7062 + 1.775.7072 + 1.775.70827
9.459.406.049.54913
16 1.751.4962 + 1.751.4972 + 1.751.49827
PRIME   9.203.225.223.02913
15 1.328.1202 + 1.328.1212 + 1.328.12227
5.291.716.171.92513
14 378.8112 + 378.8122 + 378.81326
430.495.594.03412
13 173.7362 + 173.7372 + 173.73826
90.553.635.50911
12 119.8112 + 119.8122 + 119.81326
43.064.746.03411
11 40.4412 + 40.4422 + 40.44325
4.906.666.09410
10 36.9972 + 36.9982 + 36.99925
4.106.556.01410
9 17.9322 + 17.9332 + 17.93425
964.777.4699
8 17.5522 + 17.5532 + 17.55425
924.323.4299
7 16.0842 + 16.0852 + 16.08625
776.181.6779
6 8.3392 + 8.3402 + 8.34124
208.666.8029
5 1.7362 + 1.7372 + 1.73824
9.051.5097
4 5662 + 5672 + 56823
964.4696
3 372 + 382 + 3922
4.3344
2 112 + 122 + 1322
4343
1 42 + 52 + 621
772
   
Sums of Squares of TWO Consecutive Integers
Palindromes are of odd length and nature only. Searched upto length 31.
Carlos Rivera collects these palindromes (Palprimes and sums of powers)
especially when they are prime (see highlights).
Submissions from Jean Claude Rosa (email)
Index 26 [ September 14, 2002 ]
Index 27 [ October 11, 2002 ]
Index 28, 29, 30 & 31 [ October 17, 18, 19 & 21, 2002 ]
Index 32 [ February 10, 2003 ]
Index 33 & 34 [ February 26, 2003 ]
Index 35 & 36 [ June 8, 2005 ]
Index 37 [ July 5, 2005 ]
Index 38 & 39 [ August 1, 2005 ]
Index 40 [ September 6, 2005 ]
Index 41 [ September 15, 2006 ]
Index 42 [ October 14, 2006 ]
Index 43 [ May 21, 2007 ]
Index 44 [ January 12, 2008 ]
Index 45 [ February 17, 2009 ]
45 72.421.835.667.809.2002 + 72.421.835.667.809.201217
10.489.844.562.990.321.812.309.926.544.898.40135
44 17.149.147.897.016.1812 + 17.149.147.897.016.182217
588.186.547.187.469.040.964.781.745.681.88533
43 16.767.809.460.615.5112 + 16.767.809.460.615.512217
562.318.868.215.014.101.410.512.868.813.26533
42 15.831.350.305.118.4762 + 15.831.350.305.118.477217
501.263.304.966.749.757.947.669.403.362.10533
41 12.577.180.410.882.0822 + 12.577.180.410.882.083217
316.370.934.175.751.979.157.571.439.073.61333
40 1.650.889.370.330.0162 + 1.650.889.370.330.017216
5.450.871.426.137.276.727.316.241.780.54531
39 1.258.375.325.735.8822 + 1.258.375.325.735.883216
3.167.016.920.841.776.771.480.296.107.61331
38 917.076.696.729.4692 + 917.076.696.729.470215
1.682.059.335.368.470.748.635.339.502.86131
37 712.254.882.551.4252 + 712.254.882.551.426215
1.014.614.035.436.689.866.345.304.164.10131
36 86.505.526.088.5702 + 86.505.526.088.571214
14.966.412.087.720.702.778.021.466.94129
35 73.675.999.227.0992 + 73.675.999.227.100214
10.856.305.724.223.132.242.750.365.80129
34 12.615.243.893.5622 + 12.615.243.893.563214
318.288.756.988.131.889.657.882.81327
33 12.613.810.689.7622 + 12.613.810.689.763214
318.216.440.234.333.432.044.612.81327
32 7.365.154.696.7752 + 7.365.154.696.776213
108.491.007.414.868.414.700.194.80127
31 1.589.811.006.1232 + 1.589.811.006.124213
5.054.998.070.382.830.708.994.50525
30 1.258.837.943.7172 + 1.258.837.943.718213
3.169.345.937.085.807.395.439.61325
29 1.255.589.965.8522 + 1.255.589.965.853213
3.153.012.324.698.964.232.103.51325
28 1.250.999.800.0122 + 1.250.999.800.013213
3.130.000.999.262.629.990.000.31325
27 844.706.005.2202 + 844.706.005.221212
1.427.056.470.511.150.746.507.24125
26 165.058.650.6662 + 165.058.650.667212
54.488.716.319.691.361.788.44523
25 16.794.058.7112 + 16.794.058.712211
564.080.816.010.618.080.46521
24 12.574.461.6172 + 12.574.461.618211
316.234.169.939.961.432.61321
23 8.055.329.3602 + 8.055.329.361210
129.776.662.212.266.677.92121
22 8.032.814.1392 + 8.032.814.140210
129.052.205.999.502.250.92121
21 7.360.311.9002 + 7.360.311.901210
108.348.382.545.283.843.80121
20 1.705.387.6432 + 1.705.387.644210
5.816.694.029.204.966.18519
19 1.680.689.7882 + 1.680.689.789210
5.649.436.330.336.349.46519
18 1.616.689.8032 + 1.616.689.804210
5.227.371.841.481.737.22519
17 80.472.2642 + 80.472.26528
12.951.570.707.515.92117
16 17.070.7062 + 17.070.70728
582.818.040.818.28515
15 12.458.5972 + 12.458.59828
310.433.303.334.01315
14 8.659.9292 + 8.659.93027
149.988.757.889.94115
13 7.901.0142 + 7.901.01527
124.852.060.258.42115
12 1.258.1822 + 1.258.18327
3.166.046.406.61313
11 904.2802 + 904.28126
1.635.446.445.36113
10 170.7062 + 170.70726
58.281.418.28511
9 1.7062 + 1.70724
5.824.2857
8 1.6212 + 1.62224
5.258.5257
7 1.2622 + 1.26324
PRIME   3.187.8137
6 1.2572 + 1.25824
3.162.6137
5 9192 + 92023
1.690.9617
4 162 + 1722
5453
3 122 + 1322
3133
2 92 + 1021 - 2
1813
1 12 + 221
51


Contributions

Jean-Claude Rosa - go to entry sum of 3 squares - go to entry sum of 2 squares

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(All rights reserved) - Last modified : May 31, 2009.
Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com